Numerical Solutions of Variable-Coefficient Fractional-in-Space KdV Equation with the Caputo Fractional Derivative

In this paper, numerical solutions of the variable-coefficient Korteweg-De Vries (vcKdV) equation with space described by the Caputo fractional derivative operator is developed. The propagation and interaction of vcKdV equation in different cases, such as breather soliton and periodic suppression soliton, are numerically simulated. Especially, the Fourier spectral method is used to solve the fractional-in-space vcKdV equation with breather soliton. From numerical simulations and compared with other methods, it can be easily seen that our method has low computational complexity and higher precision.


Introduction
The KdV equation is one of the most notable integrable equations and has found numerous applications in many fields of science, such as plasma physics, nonlinear optics, telecommunications, fluid mechanics, condensed matter physics and dust plasma. The KdV equation was derived by Korteweg and de Vires in 1895 [1]. There are many analytical methods to obtain the analytical solutions of the KdV equation, including Hirota method, Darboux transformation and so on [2,3]. The symmetry method is used to solve many fractional differential equations, such as the seventh-order generalized KdV equation [4], the generalized KdV-Burgers-Kuramoto equation [5] and the time fractional generalized fifth-order KdV equation [6]. Qin [7] used the Hirota method to obtain the N-soliton solutions of the coupled KdV-mKdV system based on Bell polynomials. Chen [8] used the test function method combined with the bilinear to obtain the lump solutions to the generalized variable coefficient Burgers equation.
There are also many good numerical methods to solve the KdV equation. Yan [9] used a local discontinuous Galerkin method to solve the KdV equation. Jackaman [10] advanced the design of the conservative finite element discretizations for the vectorial modified KdV equation. Energy-conserving Hamiltonian boundary value methods were used to solve the KdV equation by Brugnano [11]. However, the numerical method that can be applied to solve the space fractional vcKdV equation with breather soliton and periodic suppression soliton is seldom researched. Therefore, a Fourier spectral method is developed in this paper which has low computational complexity and higher precision. We consider the KdV equation with the following form: equipped with the following initial and boundary conditions, where D β 1 x v, D β 2 x vs. denote Caputo fractional derivative operator which has the following form, and ε, α 1 , α 2 , α 3 are positive parameters.
Consulting materials in the literature, we discover that the numerical solutions of the space fractional vcKdV equation are rarely researched. Therefore, some numerical solutions of the vcKdV equation are given in this manuscript. From the numerical simulations and compared with other methods, it can be easily seen that our method has low computational complexity and higher precision. This paper is organized as follows: In Section 2, we introduce the numerical method. Many numerical experiments are provided in Section 3. Section 4 concludes the paper.

Definition 1.
A class of the single-step method for solving an ordinary differential equation in the form of: where incremental function φ is determined by R(t, V), that is a function of V n , t n , τ.

Theorem 1.
If φ(V, t, τ) satisfies the Lipschitz condition in V, then the numerical method that is given by Equation (15) is stable.
Proof. We refer the reader to [22][23][24] for the details of the proof.

Theorem 3.
Let V(t n ) be the analytical solution of problem (13) and V n is the numerical solution.

Proof. If Equation
By denoting e n = V(t n ) − V n , we have Summing over n, we get Using the Gronwall inequality, we have Finally, we find the numerical solution using the inverse discrete Fourier transform [25].

Simulation Results
Numerical solutions of the space fractional vcKdV equation are obtained in this chapter. We use the error norms, L 2 , L ∞ and GRE (global relative error) to test the accuracy of the method: and v * (x j , t) are the numerical solution and analytical solution. The order of convergence in space is computed by where c, x 0 are constant parameters.
Referring to the numerical experiment in [26][27][28][29][30], the analytical solution is as follows: In this simulation, we set α 2 = 4.84 × 10 −4 , c = 0.5, x 0 = 6, τ = 0.01 and N = 512. By present method, the numerical results are given in Table 1. Tables 2 and 3 show the absolute error by our numerical method, Hybrid method [26], B-spline method [27], ANS method [29], HBI method [30] at t = 0.005, 0.01. Then, we shall investigate the space fractional KdV equation. Table 4 shows comparison of L ∞ at different β 2 . For the space fractional KdV equation, we take (26) as reference solution because the analytical solution can not be obtained. Figures 1 and 2 show the logarithm of absolute errors by present method, Hybrid numerical method [26], B-spline method [27], ANS method [29], HBI method [30] at the the selected notes for t = 0.005, 0.01, which shows that our numerical method has higher accuracy than other methods. Comparisons are made between numerical solutions and analytical solutions at t = 0, 1, 2 in Figure 3. Figure 4 presents absolute error at x = 2. Numerical solutions at t = 1, 2 and β 2 = 0.9, 0.99, 1 are plotted in Figures 5 and 6. Numerical solutions at different β 2 are presented in Figure 7. Table 1. Spatial numerical errors L ∞ , L 2 and their corresponding convergence rates at t = 0.001 for Example 1.           From Example 1, we know that our numerical method has higher accuracy than other methods for the one-soliton solution. Next, we will study the generalized vc KdV-mKdV equation and the influence of β 1 , β 2 on the numerical solution of this equation. ).
Wang [31] obtained one periodic depression soliton solution of the generalized vcKdV- ). Table 5 shows L ∞ and GRE at different times. Table 6 shows numerical results. Figures 8-10 represent numerical solution. Absolute errors at t = 3.4, 3.5, 3.6 are shown in Figures 11-14 present absolute errors at x = −10, −5, 10. From Figure 15 which shows the numerical solutions at different β 1 and β 2 , we can find that the change of β 1 and β 2 has minimal effect on the shape of periodic depression soliton.           From Example 2, we find that our numerical method has higher accuracy and low computational complexity than other methods for one periodic depression soliton solution of the generalized vc KdV-mKdV equation. Next, we will investigate the breather-type solution of the mKdV equation.
Case I The initial value is as follows, with where k 1 , k 2 are constant parameters.

Conclusions
In this manuscript, we study the influence of β 1 , β 2 on the numerical solutions of the spatial fractional vcKdV equation. Comparisons are made between the present method and others methods; it can be easily seen that our method has low computational complexity and higher precision. Through Examples 1-3, we know that if β 1 , β 2 tends to 1, the numerical solution of the spatial fractional vcKdV equation tends to the analytical solution of the original equation. Through Example 3, the solutions of the space fractional KdV equation are very sensitive to a change in β 1 , β 2 . From Example 2, we can find that a change in β 1 and β 2 has a minimal effect on the shape of a periodic depression soliton. These results are consistent with the numerical simulation of other scholars [24,27,32].
All computations are performed by the MatlabR2017b software. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data used to support the findings of this study are available from the corresponding author upon request.